May a woman travel to the pilgrimage without a male relative?
Being accompanied by a close relative serves to protect a woman on the journey. If women today travel long distances safely on their own anyway, there is no reason to single out the pilgrimage as the exception.
With this question it helps to distinguish two parts of the journey. The first part is the pilgrimage itself. Once the pilgrims have entered the state of consecration, an atmosphere of sacredness prevails. Everyone is turned towards God, and the verse above forbids every indecent word and every quarrel during this time. Here a woman faces no particular difficulty.
The second part is the way from one's own country to that place. For this, it used to be the rule that a woman on long journeys was accompanied by a husband, father or another close relative. The purpose of this rule was her protection, because travelling over long distances was dangerous and took weeks.
Times have changed. Routes and means of travel are no longer what they were. Women today fly alone across thousands of kilometres, and nobody sees anything wrong in that. It is then hard to explain why the journey to the pilgrimage, of all journeys, should be treated differently. Either one keeps women from travelling alone in general, or one does not single out the pilgrimage as the only exception.
That is the basic idea. In an individual case, a woman planning the pilgrimage should ask knowledgeable scholars about the exact rulings and make sure the journey is safe, for example in a trustworthy group. The duty of pilgrimage applies to women as to men once a way to it is open.